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Resolución de integrales/ 3sinx/ sinnx/ 3sinxsinnx

Integral de 3sinxsinnx dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                     
  /                     
 |                      
 |  3*sin(x)*sin(n*x) dx
 |                      
/                       
0
$$\int\limits_{0}^{1} 3 \sin{\left(x \right)} \sin{\left(n x \right)}\, dx$$ 
Integral((3*sin(x))*sin(n*x), (x, 0, 1))
Respuesta (Indefinida) [src] 
                              //         2             2                                 \
                              ||  3*x*cos (x)   3*x*sin (x)   3*cos(x)*sin(x)            |
                              ||- ----------- - ----------- + ---------------  for n = -1|
                              ||       2             2               2                   |
                              ||                                                         |
  /                           ||                           2             2               |
 |                            ||  3*cos(x)*sin(x)   3*x*cos (x)   3*x*sin (x)            |
 | 3*sin(x)*sin(n*x) dx = C + |<- --------------- + ----------- + -----------  for n = 1 |
 |                            ||         2               2             2                 |
/                             ||                                                         |
                              ||   3*cos(x)*sin(n*x)   3*n*cos(n*x)*sin(x)               |
                              ||   ----------------- - -------------------     otherwise |
                              ||              2                    2                     |
                              ||        -1 + n               -1 + n                      |
                              \\                                                         /
$$\int 3 \sin{\left(x \right)} \sin{\left(n x \right)}\, dx = C + \begin{cases} - \frac{3 x \sin^{2}{\left(x \right)}}{2} - \frac{3 x \cos^{2}{\left(x \right)}}{2} + \frac{3 \sin{\left(x \right)} \cos{\left(x \right)}}{2} & \text{for}\: n = -1 \\\frac{3 x \sin^{2}{\left(x \right)}}{2} + \frac{3 x \cos^{2}{\left(x \right)}}{2} - \frac{3 \sin{\left(x \right)} \cos{\left(x \right)}}{2} & \text{for}\: n = 1 \\- \frac{3 n \sin{\left(x \right)} \cos{\left(n x \right)}}{n^{2} - 1} + \frac{3 \sin{\left(n x \right)} \cos{\left(x \right)}}{n^{2} - 1} & \text{otherwise} \end{cases}$$
Simplificar
Respuesta [src] 
/       2           2                                 
|  3*cos (1)   3*sin (1)   3*cos(1)*sin(1)            
|- --------- - --------- + ---------------  for n = -1
|      2           2              2                   
|                                                     
|      2           2                                  
| 3*cos (1)   3*sin (1)   3*cos(1)*sin(1)             
< --------- + --------- - ---------------   for n = 1 
|     2           2              2                    
|                                                     
|   3*cos(1)*sin(n)   3*n*cos(n)*sin(1)               
|   --------------- - -----------------     otherwise 
|             2                  2                    
|       -1 + n             -1 + n                     
\
$$\begin{cases} - \frac{3 \sin^{2}{\left(1 \right)}}{2} - \frac{3 \cos^{2}{\left(1 \right)}}{2} + \frac{3 \sin{\left(1 \right)} \cos{\left(1 \right)}}{2} & \text{for}\: n = -1 \\- \frac{3 \sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{3 \cos^{2}{\left(1 \right)}}{2} + \frac{3 \sin^{2}{\left(1 \right)}}{2} & \text{for}\: n = 1 \\- \frac{3 n \sin{\left(1 \right)} \cos{\left(n \right)}}{n^{2} - 1} + \frac{3 \sin{\left(n \right)} \cos{\left(1 \right)}}{n^{2} - 1} & \text{otherwise} \end{cases}$$ 
=
= 
/       2           2                                 
|  3*cos (1)   3*sin (1)   3*cos(1)*sin(1)            
|- --------- - --------- + ---------------  for n = -1
|      2           2              2                   
|                                                     
|      2           2                                  
| 3*cos (1)   3*sin (1)   3*cos(1)*sin(1)             
< --------- + --------- - ---------------   for n = 1 
|     2           2              2                    
|                                                     
|   3*cos(1)*sin(n)   3*n*cos(n)*sin(1)               
|   --------------- - -----------------     otherwise 
|             2                  2                    
|       -1 + n             -1 + n                     
\
$$\begin{cases} - \frac{3 \sin^{2}{\left(1 \right)}}{2} - \frac{3 \cos^{2}{\left(1 \right)}}{2} + \frac{3 \sin{\left(1 \right)} \cos{\left(1 \right)}}{2} & \text{for}\: n = -1 \\- \frac{3 \sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{3 \cos^{2}{\left(1 \right)}}{2} + \frac{3 \sin^{2}{\left(1 \right)}}{2} & \text{for}\: n = 1 \\- \frac{3 n \sin{\left(1 \right)} \cos{\left(n \right)}}{n^{2} - 1} + \frac{3 \sin{\left(n \right)} \cos{\left(1 \right)}}{n^{2} - 1} & \text{otherwise} \end{cases}$$ 
Piecewise((-3*cos(1)^2/2 - 3*sin(1)^2/2 + 3*cos(1)*sin(1)/2, n = -1), (3*cos(1)^2/2 + 3*sin(1)^2/2 - 3*cos(1)*sin(1)/2, n = 1), (3*cos(1)*sin(n)/(-1 + n^2) - 3*n*cos(n)*sin(1)/(-1 + n^2), True))

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

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